Lecture 3

22.01.2019

Phase space, Liouville’s theorem,

statistical ensembles
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Statistical mechanics
• Looks at macroscopic properties of matter from a microscopic, particle-
based point of view
• Contains two key elements:
• Particle mechanics: Laws of motion of the individual particles

• Statistical ensembles: Collections of all accessible microscopic configurations of

N particles corresponding to a macroscopic (thermodynamic) equilibrium state of
the system

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Equilibrium statistical mechanics
• Particle mechanics: Hamiltonian particle dynamics
#$
Hamiltonian of one particle ! = +( )
%&
Newton’s law of motion
+, +,
)̇ = , -̇ = −
+# +/

• Statistical ensembles: Each configuration of particle is a representative point in the

space of all particle coordinates, called phase space 01 , 31 4567518.
We define a density of states :(-, )), such that : -, ) = >? -= >? : is the number of
configurations in a volume element dA ≡ = >? -= >? : around a particular point -, ) in the phase
space

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 Phase space example: 1d Harmonic Oscillator
• Total energy .

./ 1 / / / T ., 6 U.U6
,= + 1 45 6
21 2

89 89 :
6̇ = , .̇ = − 8< → 6̇ = > , .̇ = −145/6 6
8:

Ä + BDC A = C

• Solution: 6 E = F cos 45E + I , . E = 1 6̇ = −Am4 sin 45E + I

:M N
• Motion at constant total energy , = + 1/45/6 / = OPQRE is on an ellipse
/> /

1
6/ + / / p/ = A/
1 45

• 2D-Phase space: T ., 6 U.U6 is the number of density of states around a point ., 6

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 Phase space
• System of N particles: each particle has a position coordinate !" and a momentum coordinate #" ,
where n = 1, ⋯ , ) labels each particle
• 3D: !" = (+" , ," , -" ) and #/ = (+"̇ , ,̇" , -"̇ ). Each particle has 22 = 6 degrees of freedom (d.o.f.)

• General coordinate for the N-collection of Hamiltonian particles

#, ! = (#4 , … , #67 , !4 , ⋯ , !67 )

• The system is described by a collection of particles with Hamiltonian dynamics

67
#;
8=9 + > !4 , ⋯ , !67
2<
/:4

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 Phase space 63 − /567895:8;<

• !, # = (!&, … , !() , #&, ⋯ , #() )

!
• !, # has N×2/ coordinates (# 0 , ! 0 )
• A state of the N particles that specify the position and
momentum of each particles is given by a
representative point in the phase space !, #
Ø Macroscopic properties of the system are determined
as ensemble averages over the density of state 01

#
ØThe macroscopic evolution of the system is described
by the ”flow” of the density of states in the phase
space

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 Ensemble density of states
• Imagine that the phase space is filled with accessible microstates in
analogy to how a container is filled with a fluid #

!(#, %)
• The ”fluid” is the large collection of identical systems that are in the
same macroscopic state are in different possible microstates
*+

• This ensemble is described by a density of systems occuping a point

in the phase space (”fluid density”) !(#, %)

• Number of systems which occupy the microstates between (#, %) %

and (# + (#, % + (%) is

! #, % ()

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 Ensemble density of states
• Ensemble density !(#, %) is the probability density of finding the system in
state (#, %) #

∫ ! #, % () = 1 !(#, %)

89
• Macroscopic thermodynamic variables are determined as averages over
!(#, %)
• E.g. internal energy
, = - #, % = ∫ ! #, % - #, % ()
%
In general

6 (7) = ∫ ! #, %, 7 6 #, % ()

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 Liouville’s theorem
• Arbitrary volume ! with the enclosing surface ". The rate at which the number of systems (representative points)
increases with time is
#
% '! (
#$ &
• The net rate at which the representative points flow across the boundary "
% '" ( * ⋅ , = % '! ∇ ⋅ ((*)
) &
E
• Conservation of the number of representative points (no sinks or sources): ∫ '! ( = − ∫& '! ∇ ⋅ ((*)
EF &

Equation of contintuity

#(
+ M ⋅ (* = 0, P = R,̇ Ṡ = (RṪ , RU̇ , ⋯ , Ṙ WX , SṪ , SU̇ , ⋯ , ṠWX )
#$
• Use the Hamiltonian eq. of motion

WX
#( #( #(
+] Ṙ ^ + Ṡ = 0
#$ #R^ #S^ ^
^_T

E` E` Ed E` Ed E`
+ ∑WX
^_T − =0→ + (, g = 0 Liouville’s theorem
EF Ebc Eec Eec Ebc EF
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 Liouville’s theorem for equilibrium systems
• For systems in thermodynamic equilibrium, all the
averages are time-independent, hence the density of = @A B
states is time-independent >C

• Liouville’s equation implies then that

!, # = %
= @A B
>?

• General solution of ensemble density commutes with

the Hamiltonian

! = !(#)

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 Liouville’s theorem and ergoticity
• Ensemble density is time-independent
= ;< !
!, # = 0 >@

• Particular solution
! = 12345.

• It means that all the available states in the phase space are equally probably for the
systems in the ensemble = ;< !
>?
• This ensemble is called microcanonical ensemble and the states are called
microstates
• Systems in the microcanonical ensemble are ergotic

7 8, 9 = ∫ ! 8, 9 7 8, 9 ;< (ensemble average)

= time-average of F

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 Statistical Equilibrium Ensembles
• Microcanonical ensemble ! ", $ ∼ &'()*.
• Describes a system at a fixed energy, volume and number of particles
• Each possible state at fixed , and N has an equal probability
/ ",$
• Canonical ensemble. ! ", $ ∼ -. 01
• describes a system at a fixed volume and number of particles, and that is thermal equilibrium with a heat bath
at a fixed temperature T
• The energy fluctuates according to a probability distribution function (PDF) P(E) determined by 2(4, 5)
• Internal energy U of the thermodynamic system is fixed by T and determined as an average , = ⟨9⟩
/ ",$ <(
• Grand canonical ensemble ! ", $, ( ∼ -. 01 ;01
• describes a system with varying number of particles and that is in thermal and chemical equilibrium with a
thermodynamic reservoir, i.e. fixed = and >
• Particle number and energy are fluctuating variables drawn from corresponding PDFs P(E), P(n)
• The average energy and number of particles are fixed by the temperature and chemical potential

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Microcanonical ensemble ! ", $ ∼ &'()*
• Describes a system at a fixed energy +, volume , and number of particles -

• Each possible state at fixed + and - has an equal probability

2
. /, 0 = 4(6 /, 0 − +),
3

with the proportionality constant fixed by the normalization condition of the ensemble
density ∫ . /, 0 :; = 1

Microcanonical density of states Σ +, ,, -

> ? = ∫ @AB C ", $ − ?

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Microcanonical ensemble
• Microcanonical density of states Σ ", $, %

Σ " = ∫ ()* + ,, - − "

• Phase space volume with energy less than or equal to D,

E D =F KL
G H,I JD

M N
Σ " = Ω(") and Ω " = ∫S (T Σ T
MN

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Correspondence to the thermodynamics
Phase space volume with energy less than or equal to 6,

Ω 6 =: @A
; <,= >?

Ω B ∼ multiplicity of a microstate, except is dimensional.

DGH IDGH J
If we rescale DE ≡
KLℏ H

1
Ω 6 := R
: @ SR T@ SR U
2Qℏ ; <,= >?

• Boltzmann Entropy
Y(6) = \ ln Ω (6)

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Thermodynamics in the microcanonical ensemble
Boltzmann Entropy
.(0, 2, 3) = 6 ln Ω (0, 2, 3)

; =>
Temperature <
= =? @,A

D =>
Pressure <
= =@ ?,A

I =>
Chemical potential <
=− =A ?,@

Helmholtz free energy N = 0 − O. = 0 − 6O ln Ω

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Microcanonical ensemble examples:
1D harmonic oscillator 2

Y 2, ( O2OY
"̈ + $&% " = %

• Solution: ( ) = * cos ./ ) + 0 , 2 ) = 3 (̇ = −Am./ sin ./ ) + 0 (

:; ?;
• Motion at constant total energy + = 1 is on an ellipse
<=> <>/=AB;

• Phase space volume of a shell corresponding to an energy gap Δ:

1 1
1 2Q R + Δ 2Q R − Δ 2QΔ Δ
F O2 O( = 2 − 2 = =
ℎ (>HIJ)LM :,?
I
L(>N J) ℎ./ ℎ./ ℎ./ ℏ./
< <

• «ℎ» is the minimum phase space shell volume between two consecutive trajectories of a quantum harmonic
I
oscillator with energy levels VW = <
+ X ℏ./

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1D harmonic oscillator: thermodynamics 0

H 0, 1 /0/H
• Phase space volume :
L
1 2345 " 5 1
Ω " = ( /0 /1 = =
2&ℏ ) *,, -. 2&ℏ ℏ67

• Entropy:
"
S = k ln (Ω) → ? " = @ ln 0G 1G
ℏ67
+ =1
2I" 2"/I67G
• Temperature:
2 = &5L
1 BS @
= = → " = @D
T BU " 2" 2&"
2 = & 2I" =
I67G 67
Equipartition of energy: an oscillator in 1D has 2 degrees of freedom (1 translational and 1 vibrational),
F
hence " D = 2 ⋅ G
@D = @D

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 Microcanonical ensemble of free particle
Hamiltonian of N non-interacting particles in 3D
.
,
!(#) = ∑*+ -
'() /0

• Microcanonical density of states

.
) *+ *+ *+ ,-
Σ 2 = 56 ∫ 8 #8 9: ∑'() −2
/3ℏ /0

/ *+
+
V *+
#'
Σ 2 = *+
∫ 8 #: ? −2
2>ℏ 2@
'()

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Microcanonical ensemble of free particle
• Equation of a hyper-sphere in a 3" −dimensional momentum space

$%& + $&& + ⋯ + $)*

&
= ,& , , = 2/0
• Surface integral over the momentum space ,
79
∫ 2 )* $ 3 ∑)*
56% &: − 0 can be evaluated from the volume of the hypersphere ;)*
8

;)* = < 2 )* $
= 7 >?

• Volume of the hyper-sphere scales with its radius ;)* = @)* ,)*
• Area of the hyper-sphere is related to the volume by

2;)*
A)*B% = = 3"@)* ,)*B%
2,

• How do we compute the constant @)* ?

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 Microcanonical ensemble of free particle
• How do we compute the constant !"# ? K

• Product of Gaussian integrals = volume integral of a Gaussian

"# =<
C "# C C
8: >?9 @ ;AB = DE = ∫ >G"# @ ;H = 3J!"# ∫ >K K"#;L @ ;H
9 ;<

D
!E = → ΩE] = DKE
Γ(z) integral: 1 !
RSTU RS
< C L < ;L L "#
∫Q >K (KE ) C @ ;H =(AVH C ) E ∫Q >? ? C @ ;A = E Γ E D "/E D "/E
!E = =
3 3 D/4
2 !
"# "# 4D "
"# 3J 3J DE DE → Ω"] =
3
K
DE = !"# Γ → !"# = =
2 2 3J 3J
Γ 2 +1 2 !

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Microcanonical ensemble of free particle
7

• Microcanonical phase space volume of free particles

V% )%
Ω " = )%
*)% , ,-. *)% = /)% 20" 1
2'ℏ
)%
%
V '1 )%
Ω " = 20" 1
2'ℏ )% 33
2 !

6Ω "
Σ " =
6"
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Entropy of the microcanonical ensemble
• Indistinguishable particles: the number of microstates is reduced by N!
Ω
Ω→
N!
• Boltzmann’s formula
*
% = ' l)
+!

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Entropy of the free particles in the microcanonical ensemble
)%
$% *
• Phase space volume Ω " = 201" 2
&'ℏ )% +,/& !

78 78;<
34 5 ∼ 34 : ∼ 34 : ∼ 34 = (:) (In high dimensions, the volume and area scales the same!)
9 9
5
S = T 34
8!
)
Z[ 2 +,
• U = VW lX Y − V lX ! − V lX W!
&'ℏ2 &

Using Stirling approx W! ∼ W log W − W

7
^ ] `: 9
S(:, ], 8) = T8 + 34
9 8 7a8ℏ9

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 Thermodynamic properties of the ideal gas
• Thermodynamic potential A
5 4 <2 @
1 2, 4, 5 = 75 + l;
2 5 3>5ℏ@
D G
BC = BF + BH + IBJ
E E
• Temperature
1 M1 M A 357 1 3
= = 75 l; 2 @ = → 2 = 57L
L M2 N,O M2 2 2 2
• Pressure
Q M1 57
= = → Q4 = 57L
L M4 R,O 4
• Sackur-Tetrode formula
H V V XY
C = SJ D + TU + TU E + TU
J W W WZℏW
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